For example, a user added an idea, other users vote for it, but the administration decides that this idea can't be implemented for some reasons and gives it a "Declined" status. The idea remains in the top and in a widget, thus taking extra place....

Obligatory registration will only discourage users from making their suggestions, thus destroying the whole idea of the project. I saw users complaining that they had to register twice - on the main web-site and on the forum.
Official reply
Now...